Method of Moments in One Line

Question

Four draws from Uniform[0, theta]: 2.1, 4.4, 0.9, 4.6. Compute the method-of-moments estimate of theta, and name one way method-of-moments can embarrass you that the MLE cannot.

Accepted Answer

Method of moments: match population moments to sample moments. For Uniform[0,theta], mathbbE[X] = theta/2, so set theta/2 = bar x: bar x = frac2.1 + 4.4 + 0.9 + 4.64 = frac124 = 3 implies hattheta_textMoM = 2bar x = mathbf6. Compare the MLE = max_i x_i = 4.6: the MoM estimate uses all the data through the mean; the MLE concentrates entirely on the extreme observation. The embarrassment. MoM can produce logically impossible estimates. If the sample were 0.5, 0.5, 0.5, 5.9, then bar x = 1.85 and hattheta_textMoM = 3.7, yet you've already observed a value of 5.9 > 3.7. The estimate contradicts the data. The MLE can never do that (the likelihood at a contradicted parameter is zero). MoM ignores everything about the distribution beyond the moments you matched. Why it survives anyway: MoM is consistent, requires no likelihood, and gives closed forms when MLE needs numerics, which is why its industrial-strength descendant, GMM (match many moment conditions, weight by their precision), is the backbone of econometric asset-pricing estimation. Calibrating a vol model by matching sample variance, or fitting a fat-tailed distribution by matching kurtosis, is method of moments in working clothe

Method of Moments in One Line

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